People often say that Grothendieck universes only serve to give "more
generality." But this is "generality" in a peculiar sense and I wish
I had a better name for it.
For example, suppose we have some theorem proved in ZFC for all
projective groups. You might be able to generalize it in the sense of
proving the same conclusion, also in ZFC, for all groups and not just
the projective. That would be a straightforward generalization.
Or you might "generalize" the theorem in the sense of extending ZFC to
include a strong inaccessible and now the theorem has more provable
instances than it did before. More projective groups provably exist
now than before, namely the inaccessible ones. But should we really
call this "generalization" of the theorem or should we call it
something else? It is actually a specialization of the ZFC axioms,
adding the requirement that inaccessibles exist.
best, Colin